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G = C24.42D4order 192 = 26·3

42nd non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.42D4, C8.24D12, D12.22D4, Dic6.22D4, M4(2).11D6, C8○D129C2, (C2×C8).72D6, C8.C47S3, C4.58(C2×D12), C8⋊D6.2C2, C4.137(S3×D4), C12.138(C2×D4), C8.D610C2, C33(D4.3D4), C12.46D44C2, C12.47D44C2, C6.51(C4⋊D4), C2.24(C12⋊D4), (C2×C24).155C22, (C2×C12).314C23, C4○D12.41C22, (C2×D12).89C22, C22.8(Q83S3), (C2×Dic6).95C22, (C3×M4(2)).8C22, C4.Dic3.39C22, (C2×C24⋊C2)⋊26C2, (C3×C8.C4)⋊8C2, (C2×C6).5(C4○D4), (C2×C4).115(C22×S3), SmallGroup(192,457)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C24.42D4
C1C3C6C12C2×C12C4○D12C8○D12 — C24.42D4
C3C6C2×C12 — C24.42D4
C1C2C2×C4C8.C4

Generators and relations for C24.42D4
 G = < a,b,c | a8=1, b12=c2=a4, bab-1=cac-1=a3, cbc-1=b11 >

Subgroups: 352 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3 [×2], C12 [×2], D6 [×3], C2×C6, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×2], C4×S3, D12, D12 [×2], C2×Dic3, C3⋊D4, C2×C12, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2 [×4], D24, Dic12, C4.Dic3, C2×C24, C3×M4(2) [×2], C2×Dic6, C2×D12, C4○D12, D4.3D4, C12.46D4, C12.47D4, C3×C8.C4, C8○D12, C2×C24⋊C2, C8⋊D6, C8.D6, C24.42D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.3D4, C12⋊D4, C24.42D4

Character table of C24.42D4

 class 12A2B2C2D34A4B4C4D6A6B8A8B8C8D8E8F8G12A12B12C24A24B24C24D24E24F24G24H
 size 112122422212242422488121222444448888
ρ1111111111111111111111111111111    trivial
ρ2111-1-1111-1-11111111-1-111111111111    linear of order 2
ρ31111-11111111-1-1-1-11-1-1111-1-1-1-111-1-1    linear of order 2
ρ4111-11111-1-111-1-1-1-1111111-1-1-1-111-1-1    linear of order 2
ρ5111-1-1111-1111-1-1-11-111111-1-1-1-1-1-111    linear of order 2
ρ6111111111-111-1-1-11-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ71111-11111-111111-1-1111111111-1-1-1-1    linear of order 2
ρ8111-11111-1111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ922200-12200-1-1-2-2-2-2200-1-1-11111-1-111    orthogonal lifted from D6
ρ1022-2002-22002-222-20000-2-2222-2-20000    orthogonal lifted from D4
ρ1122-2002-22002-2-2-220000-2-22-2-2220000    orthogonal lifted from D4
ρ1222-2-2022-2202-2000000022-200000000    orthogonal lifted from D4
ρ1322200-12200-1-1222-2-200-1-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1422-22022-2-202-2000000022-200000000    orthogonal lifted from D4
ρ1522200-12200-1-12222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1622200-12200-1-1-2-2-22-200-1-1-1111111-1-1    orthogonal lifted from D6
ρ1722-200-1-2200-1122-2000011-1-1-1113-3-33    orthogonal lifted from D12
ρ1822-200-1-2200-1122-2000011-1-1-111-333-3    orthogonal lifted from D12
ρ1922-200-1-2200-11-2-22000011-111-1-13-33-3    orthogonal lifted from D12
ρ2022-200-1-2200-11-2-22000011-111-1-1-33-33    orthogonal lifted from D12
ρ21222002-2-20022000002i-2i-2-2-200000000    complex lifted from C4○D4
ρ22222002-2-2002200000-2i2i-2-2-200000000    complex lifted from C4○D4
ρ2344-400-24-400-220000000-2-2200000000    orthogonal lifted from S3×D4
ρ2444400-2-4-400-2-2000000022200000000    orthogonal lifted from Q83S3, Schur index 2
ρ254-400040000-402-2-2-200000000-2-22-2000000    complex lifted from D4.3D4
ρ264-400040000-40-2-22-2000000002-2-2-2000000    complex lifted from D4.3D4
ρ274-4000-2000020-2-22-20000023-230--2-2-6--60000    complex faithful
ρ284-4000-20000202-2-2-200000-23230-2--2-6--60000    complex faithful
ρ294-4000-20000202-2-2-20000023-230-2--2--6-60000    complex faithful
ρ304-4000-2000020-2-22-200000-23230--2-2--6-60000    complex faithful

Smallest permutation representation of C24.42D4
On 48 points
Generators in S48
(1 31 7 37 13 43 19 25)(2 38 20 32 14 26 8 44)(3 33 9 39 15 45 21 27)(4 40 22 34 16 28 10 46)(5 35 11 41 17 47 23 29)(6 42 24 36 18 30 12 48)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 45 13 33)(2 32 14 44)(3 43 15 31)(4 30 16 42)(5 41 17 29)(6 28 18 40)(7 39 19 27)(8 26 20 38)(9 37 21 25)(10 48 22 36)(11 35 23 47)(12 46 24 34)

G:=sub<Sym(48)| (1,31,7,37,13,43,19,25)(2,38,20,32,14,26,8,44)(3,33,9,39,15,45,21,27)(4,40,22,34,16,28,10,46)(5,35,11,41,17,47,23,29)(6,42,24,36,18,30,12,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,45,13,33)(2,32,14,44)(3,43,15,31)(4,30,16,42)(5,41,17,29)(6,28,18,40)(7,39,19,27)(8,26,20,38)(9,37,21,25)(10,48,22,36)(11,35,23,47)(12,46,24,34)>;

G:=Group( (1,31,7,37,13,43,19,25)(2,38,20,32,14,26,8,44)(3,33,9,39,15,45,21,27)(4,40,22,34,16,28,10,46)(5,35,11,41,17,47,23,29)(6,42,24,36,18,30,12,48), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,45,13,33)(2,32,14,44)(3,43,15,31)(4,30,16,42)(5,41,17,29)(6,28,18,40)(7,39,19,27)(8,26,20,38)(9,37,21,25)(10,48,22,36)(11,35,23,47)(12,46,24,34) );

G=PermutationGroup([(1,31,7,37,13,43,19,25),(2,38,20,32,14,26,8,44),(3,33,9,39,15,45,21,27),(4,40,22,34,16,28,10,46),(5,35,11,41,17,47,23,29),(6,42,24,36,18,30,12,48)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,45,13,33),(2,32,14,44),(3,43,15,31),(4,30,16,42),(5,41,17,29),(6,28,18,40),(7,39,19,27),(8,26,20,38),(9,37,21,25),(10,48,22,36),(11,35,23,47),(12,46,24,34)])

Matrix representation of C24.42D4 in GL4(𝔽73) generated by

4862530
11375353
004811
006237
,
14661110
77111
014667
5906659
,
25116761
36486167
002562
003748
G:=sub<GL(4,GF(73))| [48,11,0,0,62,37,0,0,53,53,48,62,0,53,11,37],[14,7,0,59,66,7,14,0,11,1,66,66,10,11,7,59],[25,36,0,0,11,48,0,0,67,61,25,37,61,67,62,48] >;

C24.42D4 in GAP, Magma, Sage, TeX

C_{24}._{42}D_4
% in TeX

G:=Group("C24.42D4");
// GroupNames label

G:=SmallGroup(192,457);
// by ID

G=gap.SmallGroup(192,457);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,555,58,1123,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^12=c^2=a^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=b^11>;
// generators/relations

Export

Character table of C24.42D4 in TeX

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